{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
  "title": "Dihedral and cyclic Ramsey numbers of the alternating 3-path",
  "canonical_statement": "$R_{\\mathrm{dih}}(P_3^{\\mathrm{alt}}, K_b) = R_{\\mathrm{cyc}}(P_3^{\\mathrm{alt}}, K_b) = 2b - 1$ for all $b \\in \\mathbb{N}$ — the $a = 3$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:event",
      "problem_id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-12T00:00:00.000Z",
      "title": "GitHub repo (preprint + Lean proof + Python checker)",
      "summary": "The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:attempt"
      ],
      "method_family_ids": [
        "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:method"
      ],
      "source_assertion_ids": [
        "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:attempt",
      "problem_id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
      "model": "Claude Fable 5",
      "provider": "Anthropic",
      "attempted_at": "2026-08-12T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The model produced the proof (the $\\mathrm{Dih}(3) = \\mathrm{Sym}(3)$ collapse, the Chvátal reduction, the cyclic corollary), the Lean 4 formalization, and the Python verification script autonomously. Human direction was limited to initiation and operational supervision.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Damnjanovic and Djordjevic, Computation of small reflective and dihedral Ramsey numbers (Conjecture 4.9)",
          "url": "https://arxiv.org/abs/2607.06817",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Basic, Damnjanovic, Stevanovic and Stosic (Conjecture 4.23)",
          "url": "https://arxiv.org/abs/2604.16188",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "sat/ - regenerated CNFs, DRAT certificates, witnesses and solver logs (commit 01a50c7)",
          "url": "https://github.com/ZestyWombat854/dihedral-ramsey/tree/01a50c7/sat",
          "kind": "code",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "GitHub repo (preprint + Lean proof + Python checker)",
          "url": "https://github.com/ZestyWombat854/dihedral-ramsey",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Dihedral and cyclic Ramsey numbers of the alternating 3-path",
          "url": "https://vibemathed.com/problem/dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:verification",
      "problem_id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
      "solution_event_id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Reproduced here on 13 August 2026. The Lean development builds clean (exit 0) on the pinned toolchain (v4.12.0, core only, no Mathlib), and #print axioms shows all five main theorems depending on exactly propext, Classical.choice and Quot.sound. No Lean.ofReduceBool; with comments stripped the source has zero sorry, admit, axiom declarations and native_decide, and its 23 decide calls are kernel-reduced. A naive grep says otherwise only because those words appear in the file's own docs. The Python checker runs as described: Dih(3) has order 6 and equals Sym(3), and the lower-bound witnesses hold for b = 2..8. The general upper bound is not formalized; it cites Chvatal 1977, whose arithmetic holds. The SAT claim, unconfirmed at review, was substantiated the same day at commit 01a50c7. The DRAT files were not replayed, since replaying a shipped proof is the weaker check; instead all twelve CNFs were re-solved here with CaDiCaL, every verdict matching their kissat logs - satisfiable at $n=2b-2$, unsatisfiable at $n=2b-1$, for b = 2..7. The six satisfiable instances had their witnesses re-substituted clause by clause and all satisfy, and the b = 3 legs agree with this site's own exhaustive enumeration, anchoring their encoder against an independent computation. The certificates are regenerated rather than the originals, disclosed unprompted, which costs nothing here. Still unconfirmed: no human peer review, this being a self-submission reviewed by AI agents in-pipeline.",
      "sources": [
        {
          "label": "VibeMathed: Dihedral and cyclic Ramsey numbers of the alternating 3-path",
          "url": "https://vibemathed.com/problem/dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "GitHub repo (preprint + Lean proof + Python checker)",
          "url": "https://github.com/ZestyWombat854/dihedral-ramsey",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "GitHub repo (preprint + Lean proof + Python checker)",
      "url": "https://github.com/ZestyWombat854/dihedral-ramsey",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Dihedral and cyclic Ramsey numbers of the alternating 3-path",
      "url": "https://vibemathed.com/problem/dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path:signal:watch_only",
      "problem_id": "vibemathed:dihedral-and-cyclic-ramsey-numbers-of-the-alternating-3-path",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}